Fractions and Volume

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SWBAT find the volume of rectangular prisms using fractional side lengths or filling the space with fractional units

Big Idea

Creating a conceptual understanding of how to find the volume of rectangular prisms using fractional side lengths.


10 minutes

This is going to be a review of multiplying with fractions.  Students will need to apply this knowledge to today’s learning. If students have difficulty multiplying the fractions, remind them to use the area model as a tool(SMP5: using tools strategically)  As a reminder, the area model works like this:  First draw a rectangle to show the given amount( use vertical lines).  Then in the same rectangle, draw horizontal lines to show the second fraction.  The overlapping area is the solution.  The students may have difficulty with multiplying mixed numbers.  They will have to change these in to improper fractions in order to model or multiply.

Tools:  Exploring Volume notes

Exploring Volume using Fractional Units

20 minutes

Students will be given the task to find the volume of a rectangular prism given a fractional cubic unit.

 For example, suppose each small cube had a length, width and height of ¼ inches.  What would be the volume of the whole cube?

There are two ways to work this.

Method 1:  Find the volume of 1 cube ¼ x ¼ x ¼ = 1/64 u³, then multiply this by the total number of cubes (length, width, and height) in the prism.  So, if the prism has a length of 4, width of 4, and a height of 4, the total number of cubes needed would be 64.  1/64 x 64 = 1 cm³.

Method 2:  If the length, width, and height of the prism have side lengths of 4 cm.  Then you would need to convert the side length into fourths.  4/4 x 4/4 x 4/4 = 1 cm³

You can show students one or both of these methods.  I find the second method easier to understand because you can write the fractional amount under each cube to show them how you got the numbers.


Finding Volume with Fractional Units Explained

10 minutes

The students will be watching a video from Learnzillion that shows them how to work with fractional side lengths. I chose to use this video because the site did a nice job showing how to split the cubes into fractional parts.  The video clearly explains how to find the solution. 

Let's Try It

20 minutes

The students are going to try and apply their learning to find the volume of two rectangular prisms. Students will need to figure out which fractional unit they will need (SMP1: making sense of the problem) then they will need to use either method 1 or method 2 to help them find the solution (SMP2: reasoning about the numbers and information)

In the first example, the students will need to see that they have to break their units into halves.  The side length 3 ½ will need 7 half units.  The next side length of 2 will need 4 half units and the third side length of 2 will also need 4 half units.  The students can multiply 7/2 x 4/2 to get 28/4 then multiply again by 4/2 to get 112/8.  When simplified the solution becomes  14 cm³.

In the second example, each cube is 1/5 cm in length, width, and height.  Students can use either method to find their solution.

Method 1:  1/5 x 1/5x 1/5 = 1/125cm ³.  Then they will need to multiply this number by the total amount of cubes in the prism which is 24.  1/125 x 24 = 24/125 cm³.

Method 2: If each cube has a length of 1/5, then they would find out that the length is 3/5, the width is 2/5 and the height is 4/5.  They would then multiply the numbers together = 24/ 125 cm³.

Students that struggle with this may need to know that when multiplying fractions, we multiply the numerators and multiply the denominators to get to our solution.  Using the area model for this might be too cumbersome and you will lose them in the process.  If needed, show them a simpler problem using the area model and how it turns into the algorithm for multiplying fractions.

Tools:  Let’s try it examples

Informal Assessment

15 minutes

 Thestudents will be using what they’ve learned and applying it to a real life volume problem.  This will be a good assessment of learning because the first part asks them to find volume using nice numbers.  The second part says if she fills the fish tank ¾ full, what is the volume of the water.  The students can find the solution two different ways:

First, they can find the full volume and then find ¾ of the solution.

For example:   20 x 20 x16  6400.  Then they could find ¾ of 6400 = 4800 units cubed

Secondly, they could see that the height of the tank would be ¾ of its original.  They could find ¾ of it by using a tape diagram or multiplying the height by ¾. The new volume would be 12 x 20 x20 = 4800 units cubed.

As students are working on this problem, walk around to see how they are working on this.  For students that struggle, encourage them to use a visual to help them solve (tape diagram to find fractional amounts). An extension would be to have the students find it using two different strategies.

Students will be modeling the math by showing that volume is found through multiplication. (SMP4: modeling). Additionally, in part B of this problem, students will need to recognize that ¾ of the height will need to be adjusted.  This recognition supports SMP2: Reasoning.

Tools: Informal Assessment Problem.


10 minutes

The students will be working on a comprehension menu to turn in for evidence of student learning.  The questions on the comprehension menu deal with volume.  Students should do all 4 problems and then put a start next to the problem that was easiest/they liked the most. The comprehension menu supports

SMP 1:  making sense of problems and looking for entry points

SMP2: making sense of relationships among different mathematics concepts

SMP3: justifying conclusions

SMP5: using tools strategically.

Tools:  Comprehension menu